2,292 research outputs found
Non-Chiral Logarithmic Couplings for the Virasoro Algebra
This Letter initiates the study of what we call non-chiral staggered Virasoro
modules, indecomposable modules on which two copies of the Virasoro algebra act
with the two zero-modes acting non-semisimply. This is motivated by the
"puzzle" recently reported in arXiv:1110.1327 [math-ph] involving a
non-standard measured value, meaning that the value is not familiar from chiral
studies, for the "b-parameter" (logarithmic coupling) of a c=0 bulk conformal
field theory. Here, an explanation is proposed by introducing a natural family
of bulk modules and showing that the only consistent, non-standard logarithmic
coupling that is distinguished through structure is that which was measured.
This observation is shown to persist for general central charges and a
conjecture is made for the values of certain non-chiral logarithmic couplings.Comment: 10 pages; v2: 11 pages, some modifications to introduction, added
conclusions and reference
sl(2)_{-1/2} and the Triplet Model
Conformal field theories with sl(2)_{-1/2} symmetry are studied with a view
to investigating logarithmic structures. Applying the parafermionic coset
construction to the non-logarithmic theory, a part of the structure of the
triplet model is uncovered. In particular, the coset theory is shown to admit
the triplet W-algebra as a chiral algebra. This motivates the introduction of
an augmented sl(2)_{-1/2}-theory for which the corresponding coset theory is
precisely the triplet model. This augmentation is envisaged to lead to a
precise characterisation of the "logarithmic lift" of the non-logarithmic
sl(2)_{-1/2}-theory that has been proposed by Lesage et al.Comment: 27 pages, 3 figures, 1 table; v2 added refs to vertex algebra
literature and a few comment
Fusion in Fractional Level sl^(2)-Theories with k=-1/2
The fusion rules of conformal field theories admitting an sl^(2)-symmetry at
level k=-1/2 are studied. It is shown that the fusion closes on the set of
irreducible highest weight modules and their images under spectral flow, but
not when "highest weight" is replaced with "relaxed highest weight". The fusion
of the relaxed modules, necessary for a well-defined u^(1)-coset, gives two
families of indecomposable modules on which the Virasoro zero-mode acts
non-diagonalisably. This confirms the logarithmic nature of the associated
theories. The structures of the indecomposable modules are completely
determined as staggered modules and it is shown that there are no logarithmic
couplings (beta-invariants). The relation to the fusion ring of the c=-2
triplet model and the implications for the beta gamma ghost system are briefly
discussed.Comment: 33 pages, 8 figures; v2 - added a ref and deleted a paragraph from
the conclusion
From Jack polynomials to minimal model spectra
In this note, a deep connection between free field realisations of conformal
field theories and symmetric polynomials is presented. We give a brief
introduction into the necessary prerequisites of both free field realisations
and symmetric polynomials, in particular Jack symmetric polynomials. Then we
combine these two fields to classify the irreducible representations of the
minimal model vertex operator algebras as an illuminating example of the power
of these methods. While these results on the representation theory of the
minimal models are all known, this note exploits the full power of Jack
polynomials to present significant simplifications of the original proofs in
the literature.Comment: 14 pages, corrected typos and added comment on connections to the AGT
conjecture in introduction, version to appear in J. Phys.
Logarithmic Conformal Field Theory: Beyond an Introduction
This article aims to review a selection of central topics and examples in
logarithmic conformal field theory. It begins with a pure Virasoro example,
critical percolation, then continues with a detailed exposition of symplectic
fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW
model on the Lie supergroup GL(1|1). It concludes with a general discussion of
the so-called staggered modules that give these theories their logarithmic
structure, before outlining a proposed strategy to understand more general
logarithmic conformal field theories. Throughout, the emphasis is on continuum
methods and their generalisation from the familiar rational case. In
particular, the modular properties of the characters of the spectrum play a
central role and Verlinde formulae are evaluated with the results compared to
the known fusion rules. Moreover, bulk modular invariants are constructed, the
structures of the corresponding bulk state spaces are elucidated, and a
formalism for computing correlation functions is discussed.Comment: Invited review by J Phys A for a special issue on LCFT; v2 updated
references; v3 fixed a few minor typo
Relating the Archetypes of Logarithmic Conformal Field Theory
Logarithmic conformal field theory is a rich and vibrant area of modern
mathematical physics with well-known applications to both condensed matter
theory and string theory. Our limited understanding of these theories is based
upon detailed studies of various examples that one may regard as archetypal.
These include the c=-2 triplet model, the Wess-Zumino-Witten model on SL(2;R)
at level k=-1/2, and its supergroup analogue on GL(1|1). Here, the latter model
is studied algebraically through representation theory, fusion and modular
invariance, facilitating a subsequent investigation of its cosets and extended
algebras. The results show that the archetypes of logarithmic conformal field
theory are in fact all very closely related, as are many other examples
including, in particular, the SL(2|1) models at levels 1 and -1/2. The
conclusion is then that the archetypal examples of logarithmic conformal field
theory are practically all the same, so we should not expect that their
features are in any way generic. Further archetypal examples must be sought.Comment: 37 pages, 2 figures, several diagrams; v2 added a few paragraphs and
reference
Takiff superalgebras and Conformal Field Theory
A class of non-semisimple extensions of Lie superalgebras is studied. They
are obtained by adjoining to the superalgebra its adjoint representation as an
abelian ideal. When the superalgebra is of affine Kac-Moody type, a
generalisation of Sugawara's construction is shown to give rise to a copy of
the Virasoro algebra and so, presumably, to a conformal field theory. Evidence
for this is detailed for the extension of the affinisation of the superalgebra
gl(1|1): Its highest weight irreducible modules are classified using spectral
flow, the irreducible supercharacters are computed and a continuum version of
the Verlinde formula is verified to give non-negative integer structure
coefficients. Interpreting these coefficients as those of the Grothendieck ring
of fusion, partial results on the true fusion ring and its indecomposable
structures are deduced.Comment: 25 page
Modular Transformations and Verlinde Formulae for Logarithmic -Models
The singlet algebra is a vertex operator algebra that is strongly
generated by a Virasoro field of central charge and a
single Virasoro primary field of conformal weight . Here, the
modular properties of the characters of the uncountably many simple modules of
each singlet algebra are investigated and the results used as the input to a
continuous analogue of the Verlinde formula to obtain the "fusion rules" of the
singlet modules. The effect of the failure of fusion to be exact in general is
studied at the level of Verlinde products and the rules derived are lifted to
the triplet algebras by regarding these algebras as simple current
extensions of their singlet cousins. The result is a relatively effortless
derivation of the triplet "fusion rules" that agrees with those previously
proposed in the literature.Comment: 22 pages, v2 minor changes; added ref
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